# Hw04ans - STAT 408 Spring 2009 Homework #4 (due Friday,...

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STAT 408 Spring 2009 Homework #4 (due Friday, February 20, by 3:00 p.m.) Be sure to show all your work; your partial credit might depend on it. No credit will be given without supporting work. 1. Sally sells seashells by the seashore. The daily sales X of the seashells have the following probability distribution: x f ( x ) x f ( x ) ( x - μ ) 2 f ( x ) 0 1 2 3 4 0.30 0.25 0.20 0.15 0.10 0.00 0.25 0.40 0.45 0.40 0.6750 0.0625 0.0500 0.3375 0.6250 1.00 1.50 1.7500 a) Find the missing probability f ( 0 ) = P ( X = 0 ). f ( 0 ) = 1 - [0.25 + 0.20 + 0.15 + 0.10] = 1 - 0.70 = 0.30 . b) Find the probability that she sells at least three seashells on a given day. P(X 3) = P(X = 3) + P(X = 4) = 0.15 + 0.10 = 0.25 . c) Find the expected daily sales of seashells. E(X) = μ X = x x f x all ) ( = 1.50 . d) Find the standard deviation of daily sales of seashells. SD(X) = X σ = ( 29 ( 29 all 2 X μ - x x f x = 75 . 1 1.3229 .

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1. (continued) Suppose each shell sells for \$5.00. However, Sally must pay \$3.00 daily for the permit to sell shells. Let Y denote Sally’s daily profit. Then Y = 5 X – 3 . e) Find the probability that Sally’s daily profit will be at least \$10.00. [ Hint: How many shells must Sally sell to have daily profit of at least \$10.00? ] x y f ( x ) = f ( y ) 0 1 2 3 4 - \$3.00 \$2.00 \$7.00 \$12.00 \$17.00 0.30 0.25 0.20 0.15 0.10 1.00 P(Y 10) = P(X 3) = 0.25 . ( Sally’s daily profit is at least \$10.00 if and only if she sells at least 3 shells. ) f) Find Sally’s expected daily profit. μ Y = E(Y) = \$5.00 E(X) - \$3.00 = \$5.00 1.50 - \$3.00 = \$ 4.50 . ( On average, Sally sells 1.5 shells per day, her expected revenue is \$7.50. Her expected profit is \$4.50 since she has to pay \$3.00 for the permit. ) OR x y f ( x ) = f ( y ) y f ( y ) 0 1 2 3 4 - \$3.00 \$2.00 \$7.00 \$12.00 \$17.00 0.30 0.25 0.20 0.15 0.10 - 0.90 0.50 1.40 1.80 1.70 1.00 4.50 μ Y = E(Y) = y y f y all ) ( = \$ 4.50 .
g) Find the standard deviation of Sally’s daily profit. σ

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## This note was uploaded on 04/27/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Hw04ans - STAT 408 Spring 2009 Homework #4 (due Friday,...

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